Thickness-dependent Evolution of the Ferroelectric Domain in Ultrathin BiFeO3 Films below 10 nm

The scaling behavior of the ferroelectric domain in thick BiFeO3 has been reported to follow Kittel’s law both theoretically and experimentally. The law collapses at a certain thickness, known as the critical thickness. However, little experimental work focused on the ferroelectric domain evolution behavior in ultrathin BiFeO3 films below 10 nm. In this work, the BiFeO3 films with various thicknesses were prepared and observed with piezoresponse force microscopy (PFM) and the periodicity of the domain was extracted by a two-dimensional fast Fourier transform (2D-FFT). The reciprocal space mapping (RSM) analysis is consistent with PFM results demonstrating the 71° domain of the BiFeO3 films. It is confirmed that the critical thickness in BiFeO3 films is below 5 nm and the domain size decreases with decreasing thickness in accordance with Kittel’s law and a scaling exponent of 0.41 down to that thickness.


Introduction
Ferroelectrics have attracted intensive interest due to their various potential applications, such as ferroelectric random access memory utilizing spontaneous polarization [1].In ferroelectrics, the domain size variates with the thickness [2,3].The scaling behavior of the ferroelectric domain against thickness is one of the critical things to understand for the manufacturing of ferroelectric devices [4,5].The evolution behaviors of the domain size are determined by the competition between several kinds of free energies, including the depolarization energy and elastic energy from the domain and domain wall energy [3].Universally, the resulting evolution of the domain size follows Kittel's law, in which the domain width w scales with a square root of the film thickness d [6][7][8].
Among the ferroelectrics, multiferroic BiFeO3 (BFO) with room-temperature magnetoelectric properties has attracted much attention [9].The scaling behavior of domain width in the BFO films follows Kittel's law for BFO films, which has been proved by many experiments [4,5,10].The behavior of the domain patterns was predicted to deviate from Kittel's law and break down below a certain thickness called critical thickness [11].Although the first-principle-based research suggested a critical thickness below about 5 unit cells, the value of the critical thickness has not been proved in BFO films experimentally [12].
Herein, BFO thin films with various thicknesses were deposited on the GdScO3 (GSO) single-crystal substrate.The ferroelectric domain structure of the ultrathin BFO films was observed with piezoresponse force microscopy (PFM).The domain periodicity was quantitatively analyzed with Two dimensional Fast Fourier transform (2D-FFT) to reveal the thickness-dependent evolution behavior of the domain size in ultrathin BFO films.

Materials and Methods
The BFO thin films of various thicknesses ranging from 5 nm to 40 nm were prepared by pulsed laser deposition (PLD) on (110)o GSO substrate using a KrF excimer laser (wavelength 248 nm, repetition rate 6 Hz, pulse width 25 ns, and Intensity 620 mJ/cm 2 ).Both the BFO target and the GSO substrate are from HeFei Crystal & Surface Technical Material Co., Ltd.The subscripts "c" and "o" denote the different descriptions in the pseudocubic crystal system and orthorhombic crystal system, respectively.The [100]c, [010]c, [001]c orientation is equal to [1][2][3][4][5][6][7][8][9][10]o, [001]o, [110]o, respectively [13].The GSO substrate has an orthorhombic structure with ao = 5.488 Å, bo = 5.746 Å, and co = 7.934 Å which can also be described as a pseudocubic structure.The transformation of the two notations follows the relationship below: ac = cc = (ao 2 + bo 2 ) 1/2 /2, bc = co/2.The ac and cc are not perpendicular and the angle of the two axes is α = 2arctan(bo/ao).Therefore, the lattice parameter of the GSO substrate in pseudocubic notation is ac = cc = 3.973 Å, bc = 3.967 Å, and α = 92.7°.The in-plane lattice parameters are slightly larger than that in bulk BFO (a = b = 3.965 Å) [14,15].The thickness of the BFO films was controlled by the pulse number under a constant deposition condition.The samples were named BFO_x, in which x represents the thickness of the thin films in the nm unit.
To assess the thickness of the films, X-ray reflectometry (XRR) measurements were performed on a high-resolution X-ray diffraction instrument (HRXRD, Malvern Panalytical Empyrean) with an incident beam of Cu Kα1 (wavelength 1.5406 Å).Theta-2theta scans and reciprocal space map (RSM) measurements were conducted on the same instrument to investigate the crystal structure of the films.Piezoresponse force microscopy (PFM) was carried out on Bruker Dimension Icon scanning probe microscope using Pt-coated conductive tips to characterize topography and the ferroelectric domain structure of the BFO thin films.The 2D-FFT for quantitative analysis of the PFM results was done on the open-source software Gwyddion.

Results and Discussion
The in-plane (IP) lattice parameters of the GSO substrate are larger than the BFO.As a result, the tensile strain caused by the lattice mismatch between BFO and GSO substrate leads to a smaller out-of-theplane (OP) lattice parameter of BFO epitaxial films on GSO than that in bulk BFO induced by Poisson's effect [16].The XRD results of all the BFO films are shown in Figure 1.The XRD profile of the GSO substrate was also measured for comparison.The BFO peaks are on the right of the substrate peak due to the larger OP parameters.The peak position of the BFO films with various thicknesses shifts from 46.225° to 45.975° as indicated in the enlarged profiles in the inset.The corresponding decrease of the OP lattice parameter from 3.925 Å to 3.945 Å indicates a release of the lattice strain with the increase in thickness.
The RSM measurement was carried out to understand the structure of the samples.The (-103)c and (013)c reflections of the BFO thin films are mapped near the (33-2)o and (420)o peaks of the substrate respectively according to the transform between the orthorhombic and pseudocubic indexes.As evident in Figure 2a and c, both of the (-103)c reflections of BFO_20 and BFO_40 show only one film peak, indicating that there is only one structural variant along the [1-10]o direction.By the way, the film peaks in (-103)c reflections become more concentrated with the increase in thickness owing to the increasing lattice relaxation with the increasing thickness.On the contrary, both of the (013)c reflections of BFO_20 and BFO_40 show two split peaks on the same side of the substrate peaks which proves the existence of two structural variants along the [001]o direction as shown in Figure 2b and d.Furthermore, the two peaks have the same Qz which means the same structural variant along the [110]o direction or the OP direction.However, it could be noted that the film peak disappears in the (013)c reflections of BFO films below 10 nm but still exists in the (-103)c reflections.It is argued that the ultra-small thickness of the thin films leads to the disappearance of the peak due to the weak diffraction intensity.PFM was taken to investigate the surface topography, domain morphology, and domain periodicities [17].The average surface roughnesses Ra of all the BFO films are lower than 300 pm, indicating a high quality of the as-grown BFO thin films.The PFM results of BFO_5, BFO_8, BFO_20, and BFO_40 are shown in Figure 3.The phase range of the phase images has been calibrated to a range from -180° to +180°.The two color contrasts in the IP PFM image correspond to the two IP polarization components, while the OP PFM of the samples displays only a single contrast, indicating a single polarization component along the OP direction.This is also confirmed by the phase distribution histogram in Figure 3(i-l), in which the IP distributions display two peaks at different phase values and the OP distributions show only one peak.It is supposed to be 71° domain on the films with only one OP polarization component and two IP polarization components according to these PFM results, consistent with the result from the RSM analysis.The results are similar in other samples (not shown here).The IP PFM phase image shows stripe-like domain configurations in all the samples, despite some break of the stripe induced by defects.A significant decrease in the domain periodicity with the thickness decrease can be directly seen in the PFM results.
The evolution behavior of the domain periodicity was analyzed quantitatively in ultrathin BFO films with various thicknesses.A typical analysis process is described here using the BFO_10 as an example for extracting the domain periodicity from the IP PFM.The IP and OP PFM phase images of the BFO_10 samples with a scan size of 500 nm are shown in Figure 4a and b.The IP PFM shows a periodic stripelike domain structure with an average domain width of about 20 nm corresponding to a periodicity of 40 nm.The two-dimensional Fourier transform (FFT) was done to the IP PFM to evaluate the domain periodicity as reported [4,18].The FFT of BFO_10 IP PFM was shown in Figure 4c.The two symmetry spots around the origin of the reciprocal space are the frequency signals originating from the domain periodic patterns.The periodicity was calculated as the reciprocal of the distance between the spot center and the origin of the reciprocal space.The mean intensity profile of the FFT along the arrow direction was extracted from the FFT image as shown in Figure 4d (black line).The center of the spot was determined by the Gaussian fit of the corresponding peak (blue line) in the profile.In BFO_10, the reciprocal distance is fitted to be 0.0225 nm -1 , which corresponds to a periodicity of 44.4 nm consistent with the value read from the PFM result directly.Similar processes were carried out for each sample to extract the periodicities.The domain periodicities are plotted in Figure 5 (red dots with error bars).The domain periodicities of the BFO thin films decrease from about 78 nm to 17 nm with the decrease of the thickness of the films from 40 nm to 5 nm.The domain periodicities are the average values extracted from several points on the samples.The data can be fitted with Kittel's law expression: w = Ad γ .The coefficients are fitted to be: A = 17.25 and scaling exponent γ = 0.41 as shown in Figure 5.The coefficient of determination R-square is 0.954 in this fitting.The findings confirm that Kittel's law was followed down to the thickness of 5 nm, which to some extent, supports the above first-principle-based theoretical calculation which claims that Kittel's law is obeyed until the thickness of 5 unit cells [12].In BFO films, the equilibrium domain configurations are also determined by the free energy of the whole system including those from domains and domain walls [3].The former consisting of depolarization energy Fd and elastic energy Fe are proportional to the domain size w.The latter consisting of the domain wall energy Fw is proportional to the reciprocal of the domain size 1/w.The equilibrium condition requests a minimum energy for the system, from which the domain width w can be calculated to have a square root relationship with the film thickness d (with a scaling exponent of γ = 0.5).The splitting of the domain leads to decreasing depolarization energy Fd and increasing domain wall energy Fw.Therefore, domain splitting will happen and lead to a smaller domain width as depolarization energy increases [19].In this work, the depolarization energy cannot be compensated by free charges because of the absence of the bottom electrode.As a result, the domain split to reach the minimum total energy and the domain width decreased.This effect becomes more significant in ultrathin films with thicknesses below 10 nm because the elastic energy and the domain wall energy approach zero when the thickness approaches zero and their contribution to the total energy can be neglected.The depolarization energy dominates the system energy in this situation.This can explain the deviation of the scaling exponent from the conventional one (0.5) and the deviation of the BFO_5 from the fitting curve.

Conclusion
In summary, the thickness-dependent evolution of the ferroelectric domain periodicity was investigated in BFO thin films.The domain periodicity of BFO thin films scale following Kittel's law with a scaling exponent of about 0.41.Kittel's law was proved to keep its validity until the film thickness of about 5 nm.The evolution of the domain width against the thickness is consistent with the theoretical prediction.
The study improves the understanding of the domain behavior in ferroelectrics which facilitates the fabrication of the ferroelectric device in the future.

Figure 1 .
Figure 1.XRD results of BFO films with various thicknesses grown on (110)o GSO substrate and that from the bared substrate.The inset is the enlarged profile of the XRD of BFO films with various thicknesses near the BFO (002)c peaks.The arrows refer to the potential BFO film peaks which were covered up by the GSO (220)o peaks.

Figure 2 .
Figure 2. The RSM results of BFO films for different crystal orientations.(a), (c), and (e) are the RSM of BFO films with different thicknesses around (-103)c reflections.(b), (d), and (f) are the ones around (013)c reflections.

Figure 3 .
Figure 3. (a-d) IP PFM phase image of BFO films with various thicknesses.(e-h) OP PFM phase image of the BFO films.(i-l) The phase distribution in both IP and OP PFM of the BFO films.The blue one is the distribution of IP PFM and the orange one is that of OP PFM.

Figure 4 .
Figure 4. (a) OP PFM phase image of BFO_10 samples with a thickness of 10 nm grown on the (110) GSO substrate.(b) IP PFM phase image of the BFO_10 samples.(c) FFT of the OP PFM phase image.(d) Mean value profile of the FFT along the white arrow direction in (c) (black line) and the Gaussian fit of the two peaks (blue line).

Figure 5 .
Figure 5. Observed domain periodicity evolution against the thickness for the (001)c-orientated BFO thin films grown on (110)o GSO substrate with 71° domain (the red points with error bar).The blue solid line is corresponding to the fitted result with a scaling exponent of 0.41 (blue line).